In recent years extensive research has been conducted on the development of different models that enable energy trading between prosumers and consumers due to expected high integration of distributed energy resources. Some of the most researched mechanisms include Peer-to-Peer energy trading, Community Self-Consumption and Transactive Energy Models. Here, we present an exhaustive review of existing research around Local Energy Market integration into distribution systems, in particular benefits and challenges that the power grid can expect from integrating these models. We also present a detailed overview of methods that are used to integrate physical network constraints into the market mechanisms, their advantages, drawbacks, and scaling potential. We find that financial energy transactions do not directly reflect the physical energy flows imposed by the constraints of the installed electrical infrastructure. Therefore, ensuring the stable and reliable delivery of electricity is vital to understand the impact of these transactions on the grid infrastructure, e.g., impacts on the control, operation, and planning of electricity distribution systems.